Now suppose that \(Y\in {\rm Gr}_{k,k+2}^{{\rm wcb}}\) , satisfies the coarse boundary
conditions and \(w_{n,k,2}\left(Y,\mathcal {Z}\right)=\left\lfloor \frac{k+1}{2}\right\rfloor \) .
This implies by the first part of the proof that the number of sign
flips \(s\)  of \(\left(\left\langle Y,1,i\right\rangle \right)_{i\in \left[n\right]}\) 
is maximal and equal to \(k\) . Hence, it follows from Theorem 5.1
of [1]} that \(Y\in \mathcal {A}_{n,k,2}\) .
